The super-approximation conjecture for finitely generated linear groups

Let $\Gamma\subset\operatorname{GL}_n(\mathbb Z)$ be a finitely generated group. For each modulus $q$, let $X_q$ be the corresponding Cayley graph of the reduction of $\Gamma$ modulo $q$, and let $T_S^{(q)}$ denote the associated averaging operator. The group $\Gamma$ has super-approximation if there exists $\epsilon>0$ such that

$$\|T_S^{(q)}\|\leq 1-\epsilon,\quad \forall q\in\mathbb N.$$

Let $\mathbb G$ be the Zariski closure of $\Gamma$; its identity component is perfect when

$$[\mathbb G^\circ,\mathbb G^\circ]=\mathbb G^\circ.$$

Super-approximation conjecture. The group $\Gamma$ has super-approximation, without restriction on the moduli, if and only if the Zariski closure $\mathbb G$ of $\Gamma$ has perfect identity component. This conjecture extends the known results for square-free moduli and prime powers, and would characterize super-approximation for all moduli in terms of the algebraic structure of the Zariski closure.